We consider the multiparameter random simplicial complex as a higher dimensional extension of the classical Erdős–Rényi graph. We investigate appearance of “unusual” topological structures in the complex from the point of view of large deviations. We first study upper tail large deviation probabilities for subcomplex counts, deriving the order of magnitude of such probabilities at the logarithmic scale precision. The obtained results are then applied to analyze large deviations for the number of simplices of the multiparameter simplicial complexes. Finally, these results are also used to deduce large deviation estimates for Betti numbers of the complex in the critical dimension.
{"title":"Large deviations for subcomplex counts and Betti numbers in multiparameter simplicial complexes","authors":"G. Samorodnitsky, Takashi Owada","doi":"10.1002/rsa.21146","DOIUrl":"https://doi.org/10.1002/rsa.21146","url":null,"abstract":"We consider the multiparameter random simplicial complex as a higher dimensional extension of the classical Erdős–Rényi graph. We investigate appearance of “unusual” topological structures in the complex from the point of view of large deviations. We first study upper tail large deviation probabilities for subcomplex counts, deriving the order of magnitude of such probabilities at the logarithmic scale precision. The obtained results are then applied to analyze large deviations for the number of simplices of the multiparameter simplicial complexes. Finally, these results are also used to deduce large deviation estimates for Betti numbers of the complex in the critical dimension.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"2 1","pages":"533 - 556"},"PeriodicalIF":1.0,"publicationDate":"2022-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80254748","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study cycle counts in permutations of 1,…,n$$ 1,dots, n $$ drawn at random according to the Mallows distribution. Under this distribution, each permutation π∈Sn$$ pi in {S}_n $$ is selected with probability proportional to qinv(π)$$ {q}^{mathrm{inv}left(pi right)} $$ , where q>0$$ q>0 $$ is a parameter and inv(π)$$ mathrm{inv}left(pi right) $$ denotes the number of inversions of π$$ pi $$ . For ℓ$$ ell $$ fixed, we study the vector (C1(Πn),…,Cℓ(Πn))$$ left({C}_1left({Pi}_nright),dots, {C}_{ell}left({Pi}_nright)right) $$ where Ci(π)$$ {C}_ileft(pi right) $$ denotes the number of cycles of length i$$ i $$ in π$$ pi $$ and Πn$$ {Pi}_n $$ is sampled according to the Mallows distribution. When q=1$$ q=1 $$ the Mallows distribution simply samples a permutation of 1,…,n$$ 1,dots, n $$ uniformly at random. A classical result going back to Kolchin and Goncharoff states that in this case, the vector of cycle counts tends in distribution to a vector of independent Poisson random variables, with means 1,12,13,…,1ℓ$$ 1,frac{1}{2},frac{1}{3},dots, frac{1}{ell } $$ . Here we show that if 01$$ q>1 $$ there is a striking difference between the behavior of the even and the odd cycles. The even cycle counts still have linear means, and when properly rescaled tend to a multivariate Gaussian distribution. For the odd cycle counts on the other hand, the limiting behavior depends on the parity of n$$ n $$ when q>1$$ q>1 $$ . Both (C1(Π2n),C3(Π2n),…)$$ left({C}_1left({Pi}_{2n}right),{C}_3left({Pi}_{2n}right),dots right) $$ and (C1(Π2n+1),C3(Π2n+1),…)$$ left({C}_1left({Pi}_{2n+1}right),{C}_3left({Pi}_{2n+1}right),dots right) $$ have discrete limiting distributions—they do not need to be renormalized—but the two limiting distributions are distinct for all q>1$$ q>1 $$ . We describe these limiting distributions in terms of Gnedin and Olshanski's bi‐infinite extension of the Mallows model. We investigate these limiting distributions further, and study the behavior of the constants involved in the Gaussian limit laws. We for example show that as q↓1$$ qdownarrow 1 $$ the expected number of 1‐cycles tends to 1/2$$ 1/2 $$ —which, curiously, differs from the value corresponding to q=1$$ q=1 $$ . In addition we exhibit an interesting “oscillating” behavior in the limiting probability measures for q>1$$ q>1 $$ and n$$ n $$ odd versus n$$ n $$ even.
{"title":"Cycles in Mallows random permutations","authors":"Jimmy He, Tobias Müller, T. Verstraaten","doi":"10.1002/rsa.21169","DOIUrl":"https://doi.org/10.1002/rsa.21169","url":null,"abstract":"We study cycle counts in permutations of 1,…,n$$ 1,dots, n $$ drawn at random according to the Mallows distribution. Under this distribution, each permutation π∈Sn$$ pi in {S}_n $$ is selected with probability proportional to qinv(π)$$ {q}^{mathrm{inv}left(pi right)} $$ , where q>0$$ q>0 $$ is a parameter and inv(π)$$ mathrm{inv}left(pi right) $$ denotes the number of inversions of π$$ pi $$ . For ℓ$$ ell $$ fixed, we study the vector (C1(Πn),…,Cℓ(Πn))$$ left({C}_1left({Pi}_nright),dots, {C}_{ell}left({Pi}_nright)right) $$ where Ci(π)$$ {C}_ileft(pi right) $$ denotes the number of cycles of length i$$ i $$ in π$$ pi $$ and Πn$$ {Pi}_n $$ is sampled according to the Mallows distribution. When q=1$$ q=1 $$ the Mallows distribution simply samples a permutation of 1,…,n$$ 1,dots, n $$ uniformly at random. A classical result going back to Kolchin and Goncharoff states that in this case, the vector of cycle counts tends in distribution to a vector of independent Poisson random variables, with means 1,12,13,…,1ℓ$$ 1,frac{1}{2},frac{1}{3},dots, frac{1}{ell } $$ . Here we show that if 01$$ q>1 $$ there is a striking difference between the behavior of the even and the odd cycles. The even cycle counts still have linear means, and when properly rescaled tend to a multivariate Gaussian distribution. For the odd cycle counts on the other hand, the limiting behavior depends on the parity of n$$ n $$ when q>1$$ q>1 $$ . Both (C1(Π2n),C3(Π2n),…)$$ left({C}_1left({Pi}_{2n}right),{C}_3left({Pi}_{2n}right),dots right) $$ and (C1(Π2n+1),C3(Π2n+1),…)$$ left({C}_1left({Pi}_{2n+1}right),{C}_3left({Pi}_{2n+1}right),dots right) $$ have discrete limiting distributions—they do not need to be renormalized—but the two limiting distributions are distinct for all q>1$$ q>1 $$ . We describe these limiting distributions in terms of Gnedin and Olshanski's bi‐infinite extension of the Mallows model. We investigate these limiting distributions further, and study the behavior of the constants involved in the Gaussian limit laws. We for example show that as q↓1$$ qdownarrow 1 $$ the expected number of 1‐cycles tends to 1/2$$ 1/2 $$ —which, curiously, differs from the value corresponding to q=1$$ q=1 $$ . In addition we exhibit an interesting “oscillating” behavior in the limiting probability measures for q>1$$ q>1 $$ and n$$ n $$ odd versus n$$ n $$ even.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"452 1","pages":"1054 - 1099"},"PeriodicalIF":1.0,"publicationDate":"2022-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82927849","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We bound the second eigenvalue of random d$$ d $$ ‐regular graphs, for a wide range of degrees d$$ d $$ , using a novel approach based on Fourier analysis. Let Gn,d$$ {G}_{n,d} $$ be a uniform random d$$ d $$ ‐regular graph on n$$ n $$ vertices, and λ(Gn,d)$$ lambda left({G}_{n,d}right) $$ be its second largest eigenvalue by absolute value. For some constant c>0$$ c>0 $$ and any degree d$$ d $$ with log10n≪d≤cn$$ {log}^{10}nll dle cn $$ , we show that λ(Gn,d)=(2+o(1))d(n−d)/n$$ lambda left({G}_{n,d}right)=left(2+o(1)right)sqrt{dleft(n-dright)/n} $$ asymptotically almost surely. Combined with earlier results that cover the case of sparse random graphs, this fully determines the asymptotic value of λ(Gn,d)$$ lambda left({G}_{n,d}right) $$ for all d≤cn$$ dle cn $$ . To achieve this, we introduce new methods that use mechanisms from discrete Fourier analysis, and combine them with existing tools and estimates on d$$ d $$ ‐regular random graphs—especially those of Liebenau and Wormald.
我们使用一种基于傅里叶分析的新方法，对随机d $$ d $$‐正则图的第二个特征值进行了绑定，用于大范围的d $$ d $$度。设Gn,d $$ {G}_{n,d} $$为n $$ n $$个顶点上的一致随机图形$$ d $$‐正则图，λ(Gn,d) $$ lambda left({G}_{n,d}right) $$为其绝对值第二大特征值。对于某常数c>0 $$ c>0 $$和任意阶d $$ d $$且log10n≪d≤cn $$ {log}^{10}nll dle cn $$，我们几乎可以肯定地证明λ(Gn,d)=(2+o(1))d(n−d)/n $$ lambda left({G}_{n,d}right)=left(2+o(1)right)sqrt{dleft(n-dright)/n} $$。结合先前涵盖稀疏随机图情况的结果，这完全确定了对于所有d≤cn $$ dle cn $$ λ(Gn,d) $$ lambda left({G}_{n,d}right) $$的渐近值。为了实现这一目标，我们引入了使用离散傅立叶分析机制的新方法，并将它们与d $$ d $$ -正则随机图(特别是Liebenau和Wormald的随机图)上的现有工具和估计相结合。
{"title":"The spectral gap of random regular graphs","authors":"Amir Sarid","doi":"10.1002/rsa.21150","DOIUrl":"https://doi.org/10.1002/rsa.21150","url":null,"abstract":"We bound the second eigenvalue of random d$$ d $$ ‐regular graphs, for a wide range of degrees d$$ d $$ , using a novel approach based on Fourier analysis. Let Gn,d$$ {G}_{n,d} $$ be a uniform random d$$ d $$ ‐regular graph on n$$ n $$ vertices, and λ(Gn,d)$$ lambda left({G}_{n,d}right) $$ be its second largest eigenvalue by absolute value. For some constant c>0$$ c>0 $$ and any degree d$$ d $$ with log10n≪d≤cn$$ {log}^{10}nll dle cn $$ , we show that λ(Gn,d)=(2+o(1))d(n−d)/n$$ lambda left({G}_{n,d}right)=left(2+o(1)right)sqrt{dleft(n-dright)/n} $$ asymptotically almost surely. Combined with earlier results that cover the case of sparse random graphs, this fully determines the asymptotic value of λ(Gn,d)$$ lambda left({G}_{n,d}right) $$ for all d≤cn$$ dle cn $$ . To achieve this, we introduce new methods that use mechanisms from discrete Fourier analysis, and combine them with existing tools and estimates on d$$ d $$ ‐regular random graphs—especially those of Liebenau and Wormald.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"24 1","pages":"557 - 587"},"PeriodicalIF":1.0,"publicationDate":"2022-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72819217","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recently, many streaming algorithms have utilized generalizations of the fact that the expected maximum distance of any 4‐wise independent random walk on a line over n steps is O(n)$$ Oleft(sqrt{n}right) $$ . In this paper, we show that 4‐wise independence is required for all of these algorithms, by constructing a 3‐wise independent random walk with expected maximum distance Ω(nlgn)$$ Omega left(sqrt{n}lg nright) $$ from the origin. We prove that this bound is tight for the first and second moment, and also extract a surprising matrix inequality from these results. Next, we consider a generalization where the steps Xi$$ {X}_i $$ are k‐wise independent random variables with bounded pth moments. We highlight the case k=4,p=2$$ k=4,p=2 $$ : here, we prove that the second moment of the furthest distance traveled is O∑Xi2$$ Oleft(sum {X}_i^2right) $$ . This implies an asymptotically stronger statement than Kolmogorov's maximal inequality that requires only 4‐wise independent random variables, and generalizes a recent result of Błasiok.
{"title":"Three‐wise independent random walks can be slightly unbounded","authors":"Shyam Narayanan","doi":"10.1002/rsa.21075","DOIUrl":"https://doi.org/10.1002/rsa.21075","url":null,"abstract":"Recently, many streaming algorithms have utilized generalizations of the fact that the expected maximum distance of any 4‐wise independent random walk on a line over n steps is O(n)$$ Oleft(sqrt{n}right) $$ . In this paper, we show that 4‐wise independence is required for all of these algorithms, by constructing a 3‐wise independent random walk with expected maximum distance Ω(nlgn)$$ Omega left(sqrt{n}lg nright) $$ from the origin. We prove that this bound is tight for the first and second moment, and also extract a surprising matrix inequality from these results. Next, we consider a generalization where the steps Xi$$ {X}_i $$ are k‐wise independent random variables with bounded pth moments. We highlight the case k=4,p=2$$ k=4,p=2 $$ : here, we prove that the second moment of the furthest distance traveled is O∑Xi2$$ Oleft(sum {X}_i^2right) $$ . This implies an asymptotically stronger statement than Kolmogorov's maximal inequality that requires only 4‐wise independent random variables, and generalizes a recent result of Błasiok.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"30 1","pages":"573 - 598"},"PeriodicalIF":1.0,"publicationDate":"2022-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75802673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the model of random Boolean expressions based on balanced binary trees with 2N$$ {2}^N $$ leaves, to which are randomly attributed one of kN$$ {k}_N $$ Boolean variables or their negations. We prove that if for every c>0$$ c>0 $$ it holds that kNexp(−cN)→0$$ {k}_Nexp left(-csqrt{N}right)to 0 $$ then asymptotically with high probability the Boolean expression is either a tautology or an antitautology. Our methods are based on the study of a certain binary operation on the set of probability measures on {0,1}I$$ {left{0,1right}}^I $$ for a finite set I.
{"title":"What is the satisfiability threshold of random balanced Boolean expressions?","authors":"Naomi Lindenstrauss, M. Talagrand","doi":"10.1002/rsa.21069","DOIUrl":"https://doi.org/10.1002/rsa.21069","url":null,"abstract":"We consider the model of random Boolean expressions based on balanced binary trees with 2N$$ {2}^N $$ leaves, to which are randomly attributed one of kN$$ {k}_N $$ Boolean variables or their negations. We prove that if for every c>0$$ c>0 $$ it holds that kNexp(−cN)→0$$ {k}_Nexp left(-csqrt{N}right)to 0 $$ then asymptotically with high probability the Boolean expression is either a tautology or an antitautology. Our methods are based on the study of a certain binary operation on the set of probability measures on {0,1}I$$ {left{0,1right}}^I $$ for a finite set I.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"56 1","pages":"599 - 615"},"PeriodicalIF":1.0,"publicationDate":"2021-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74295055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We elucidate the relationship between the threshold and the expectation‐threshold of a down‐set. Qualitatively, our main result demonstrates that there exist down‐sets with polynomial gaps between their thresholds and expectation‐thresholds; in particular, the logarithmic gap predictions of Kahn–Kalai and Talagrand (recently proved by Park–Pham and Frankston–Kahn–Narayanan–Park) about up‐sets do not apply to down‐sets. Quantitatively, we show that any collection 𝒢 of graphs on [n] that covers the family of all triangle‐free graphs on [n] satisfies the inequality ∑G∈𝒢exp(−δe(Gc)/n)<1/2 for some universal δ>0 , and this is essentially best‐possible.
{"title":"Down‐set thresholds","authors":"Benjamin Gunby, Xiaoyu He, Bhargav P. Narayanan","doi":"10.1002/rsa.21148","DOIUrl":"https://doi.org/10.1002/rsa.21148","url":null,"abstract":"We elucidate the relationship between the threshold and the expectation‐threshold of a down‐set. Qualitatively, our main result demonstrates that there exist down‐sets with polynomial gaps between their thresholds and expectation‐thresholds; in particular, the logarithmic gap predictions of Kahn–Kalai and Talagrand (recently proved by Park–Pham and Frankston–Kahn–Narayanan–Park) about up‐sets do not apply to down‐sets. Quantitatively, we show that any collection 𝒢 of graphs on [n] that covers the family of all triangle‐free graphs on [n] satisfies the inequality ∑G∈𝒢exp(−δe(Gc)/n)<1/2 for some universal δ>0 , and this is essentially best‐possible.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"1 1","pages":"442 - 456"},"PeriodicalIF":1.0,"publicationDate":"2021-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90440237","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a permutation σ$$ sigma $$ , its corresponding binary search tree is obtained by recursively inserting the values σ(1),…,σ(n)$$ sigma (1),dots, sigma (n) $$ into a binary tree so that the label of each node is larger than the labels of its left subtree and smaller than the labels of its right subtree. In this article, we study the height of binary search trees drawn from the record‐biased model of permutations whose probability measure on the set of permutations is proportional to θrecord(σ)$$ {theta}^{mathrm{record}left(sigma right)} $$ , where record(σ)=|{i∈[n]:∀jσ(j)}|$$ mathrm{record}left(sigma right)=mid left{iin left[nright]:forall jsigma (j)right}mid $$ . We show that the height of a binary search tree built from a record‐biased permutation of size n$$ n $$ with parameter θ$$ theta $$ is of order (1+oℙ(1))max{c∗logn,θlog(1+n/θ)}$$ left(1+{o}_{mathbb{P}}(1)right)max left{{c}^{ast}log n,kern0.3em theta log left(1+n/theta right)right} $$ , hence extending previous results of Devroye on the height or random binary search trees.
{"title":"The height of record‐biased trees","authors":"Benoît Corsini","doi":"10.1002/rsa.21110","DOIUrl":"https://doi.org/10.1002/rsa.21110","url":null,"abstract":"Given a permutation σ$$ sigma $$ , its corresponding binary search tree is obtained by recursively inserting the values σ(1),…,σ(n)$$ sigma (1),dots, sigma (n) $$ into a binary tree so that the label of each node is larger than the labels of its left subtree and smaller than the labels of its right subtree. In this article, we study the height of binary search trees drawn from the record‐biased model of permutations whose probability measure on the set of permutations is proportional to θrecord(σ)$$ {theta}^{mathrm{record}left(sigma right)} $$ , where record(σ)=|{i∈[n]:∀jσ(j)}|$$ mathrm{record}left(sigma right)=mid left{iin left[nright]:forall jsigma (j)right}mid $$ . We show that the height of a binary search tree built from a record‐biased permutation of size n$$ n $$ with parameter θ$$ theta $$ is of order (1+oℙ(1))max{c∗logn,θlog(1+n/θ)}$$ left(1+{o}_{mathbb{P}}(1)right)max left{{c}^{ast}log n,kern0.3em theta log left(1+n/theta right)right} $$ , hence extending previous results of Devroye on the height or random binary search trees.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"206 1","pages":"623 - 644"},"PeriodicalIF":1.0,"publicationDate":"2021-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77052046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
How efficiently can we find an unknown graph using distance queries between its vertices? We assume that the unknown graph is connected, unweighted, and has bounded degree. The goal is to find every edge in the graph. This problem admits a reconstruction algorithm based on multi‐phase Voronoi‐cell decomposition and using Õ(n3/2)$$ overset{widetilde }{O}left({n}^{3/2}right) $$ distance queries. In our work, we analyze a simple reconstruction algorithm. We show that, on random Δ$$ Delta $$ ‐regular graphs, our algorithm uses Õ(n)$$ overset{widetilde }{O}(n) $$ distance queries. As by‐products, with high probability, we can reconstruct those graphs using log2n$$ {log}^2n $$ queries to an all‐distances oracle or Õ(n)$$ overset{widetilde }{O}(n) $$ queries to a betweenness oracle, and we bound the metric dimension of those graphs by log2n$$ {log}^2n $$ . Our reconstruction algorithm has a very simple structure, and is highly parallelizable. On general graphs of bounded degree, our reconstruction algorithm has subquadratic query complexity.
{"title":"A simple algorithm for graph reconstruction","authors":"Claire Mathieu, Hang Zhou","doi":"10.1002/rsa.21143","DOIUrl":"https://doi.org/10.1002/rsa.21143","url":null,"abstract":"How efficiently can we find an unknown graph using distance queries between its vertices? We assume that the unknown graph is connected, unweighted, and has bounded degree. The goal is to find every edge in the graph. This problem admits a reconstruction algorithm based on multi‐phase Voronoi‐cell decomposition and using Õ(n3/2)$$ overset{widetilde }{O}left({n}^{3/2}right) $$ distance queries. In our work, we analyze a simple reconstruction algorithm. We show that, on random Δ$$ Delta $$ ‐regular graphs, our algorithm uses Õ(n)$$ overset{widetilde }{O}(n) $$ distance queries. As by‐products, with high probability, we can reconstruct those graphs using log2n$$ {log}^2n $$ queries to an all‐distances oracle or Õ(n)$$ overset{widetilde }{O}(n) $$ queries to a betweenness oracle, and we bound the metric dimension of those graphs by log2n$$ {log}^2n $$ . Our reconstruction algorithm has a very simple structure, and is highly parallelizable. On general graphs of bounded degree, our reconstruction algorithm has subquadratic query complexity.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"60 1","pages":"512 - 532"},"PeriodicalIF":1.0,"publicationDate":"2021-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84387331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For integers d≥2$$ dge 2 $$ and k≥d+1$$ kge d+1 $$ , a k$$ k $$‐hole in a set S$$ S $$ of points in general position in ℝd$$ {mathbb{R}}^d $$ is a k$$ k $$ ‐tuple of points from S$$ S $$ in convex position such that the interior of their convex hull does not contain any point from S$$ S $$ . For a convex body K⊆ℝd$$ Ksubseteq {mathbb{R}}^d $$ of unit d$$ d $$ ‐dimensional volume, we study the expected number EHd,kK(n)$$ E{H}_{d,k}^K(n) $$ of k$$ k $$ ‐holes in a set of n$$ n $$ points drawn uniformly and independently at random from K$$ K $$ . We prove an asymptotically tight lower bound on EHd,kK(n)$$ E{H}_{d,k}^K(n) $$ by showing that, for all fixed integers d≥2$$ dge 2 $$ and k≥d+1$$ kge d+1 $$ , the number EHd,kK(n)$$ E{H}_{d,k}^K(n) $$ is at least Ω(nd)$$ Omega left({n}^dright) $$ . For some small holes, we even determine the leading constant limn→∞n−dEHd,kK(n)$$ {lim}_{nto infty }{n}^{-d}E{H}_{d,k}^K(n) $$ exactly. We improve the currently best‐known lower bound on limn→∞n−dEHd,d+1K(n)$$ {lim}_{nto infty }{n}^{-d}E{H}_{d,d+1}^K(n) $$ by Reitzner and Temesvari (2019). In the plane, we show that the constant limn→∞n−2EH2,kK(n)$$ {lim}_{nto infty }{n}^{-2}E{H}_{2,k}^K(n) $$ is independent of K$$ K $$ for every fixed k≥3$$ kge 3 $$ and we compute it exactly for k=4$$ k=4 $$ , improving earlier estimates by Fabila‐Monroy, Huemer, and Mitsche and by the authors.
对于整数d≥2$$ dge 2 $$ k≥d+1$$ kge d+1 $$ ， a k$$ k $$‐一组中的孔$$ S $$ 在一般位置上的点$$ {mathbb{R}}^d $$ 是k吗?$$ k $$ ‐来自S的点的元组$$ S $$ 处于凸位置，使得它们的凸壳内部不包含来自S的任何点$$ S $$ ． 对于一个凸体K⊥∈d$$ Ksubseteq {mathbb{R}}^d $$ 单位d的$$ d $$ 我们研究了期望数EHd,kK(n)$$ E{H}_{d,k}^K(n) $$ k的$$ k $$ ‐一组n中的孔$$ n $$ 从K中均匀独立随机抽取的点$$ K $$ ． 我们证明了EHd,kK(n)的渐近紧下界。$$ E{H}_{d,k}^K(n) $$ 通过证明，对于所有固定整数d≥2$$ dge 2 $$ k≥d+1$$ kge d+1 $$ ，数字EHd,kK(n)$$ E{H}_{d,k}^K(n) $$ 至少是Ω(nd)$$ Omega left({n}^dright) $$ ． 对于一些小孔，我们甚至确定了前导常数limn→∞n−dEHd,kK(n)$$ {lim}_{nto infty }{n}^{-d}E{H}_{d,k}^K(n) $$ 没错。我们改进了目前已知的limn→∞n−dEHd,d+1K(n)的下界。$$ {lim}_{nto infty }{n}^{-d}E{H}_{d,d+1}^K(n) $$ 雷茨纳和特梅斯瓦里(2019)。在平面上，我们证明了常数limn→∞n−2EH2,kK(n)$$ {lim}_{nto infty }{n}^{-2}E{H}_{2,k}^K(n) $$ 与K无关$$ K $$ 对于每一个固定k≥3$$ kge 3 $$ 我们计算k=4时的结果$$ k=4 $$ ，改进了Fabila - Monroy、Huemer和Mitsche以及作者早期的估计。
{"title":"Tight bounds on the expected number of holes in random point sets","authors":"M. Balko, M. Scheucher, P. Valtr","doi":"10.1002/rsa.21088","DOIUrl":"https://doi.org/10.1002/rsa.21088","url":null,"abstract":"For integers d≥2$$ dge 2 $$ and k≥d+1$$ kge d+1 $$ , a k$$ k $$‐hole in a set S$$ S $$ of points in general position in ℝd$$ {mathbb{R}}^d $$ is a k$$ k $$ ‐tuple of points from S$$ S $$ in convex position such that the interior of their convex hull does not contain any point from S$$ S $$ . For a convex body K⊆ℝd$$ Ksubseteq {mathbb{R}}^d $$ of unit d$$ d $$ ‐dimensional volume, we study the expected number EHd,kK(n)$$ E{H}_{d,k}^K(n) $$ of k$$ k $$ ‐holes in a set of n$$ n $$ points drawn uniformly and independently at random from K$$ K $$ . We prove an asymptotically tight lower bound on EHd,kK(n)$$ E{H}_{d,k}^K(n) $$ by showing that, for all fixed integers d≥2$$ dge 2 $$ and k≥d+1$$ kge d+1 $$ , the number EHd,kK(n)$$ E{H}_{d,k}^K(n) $$ is at least Ω(nd)$$ Omega left({n}^dright) $$ . For some small holes, we even determine the leading constant limn→∞n−dEHd,kK(n)$$ {lim}_{nto infty }{n}^{-d}E{H}_{d,k}^K(n) $$ exactly. We improve the currently best‐known lower bound on limn→∞n−dEHd,d+1K(n)$$ {lim}_{nto infty }{n}^{-d}E{H}_{d,d+1}^K(n) $$ by Reitzner and Temesvari (2019). In the plane, we show that the constant limn→∞n−2EH2,kK(n)$$ {lim}_{nto infty }{n}^{-2}E{H}_{2,k}^K(n) $$ is independent of K$$ K $$ for every fixed k≥3$$ kge 3 $$ and we compute it exactly for k=4$$ k=4 $$ , improving earlier estimates by Fabila‐Monroy, Huemer, and Mitsche and by the authors.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"61 1","pages":"29 - 51"},"PeriodicalIF":1.0,"publicationDate":"2021-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83832028","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a graph G$$ G $$ of degree k$$ k $$ over n$$ n $$ vertices, we consider the problem of computing a near maximum cut or a near minimum bisection in polynomial time. For graphs of girth 2L$$ 2L $$ , we develop a local message passing algorithm whose complexity is O(nkL)$$ O(nkL) $$ , and that achieves near optimal cut values among all L$$ L $$ ‐local algorithms. Focusing on max‐cut, the algorithm constructs a cut of value nk/4+nP⋆k/4+err(n,k,L)$$ nk/4+n{P}_{star}sqrt{k/4}+mathsf{err}left(n,k,Lright) $$ , where P⋆≈0.763166$$ {P}_{star}approx 0.763166 $$ is the value of the Parisi formula from spin glass theory, and err(n,k,L)=on(n)+nok(k)+nkoL(1)$$ mathsf{err}left(n,k,Lright)={o}_n(n)+n{o}_kleft(sqrt{k}right)+nsqrt{k}{o}_L(1) $$ (subscripts indicate the asymptotic variables). Our result generalizes to locally treelike graphs, that is, graphs whose girth becomes 2L$$ 2L $$ after removing a small fraction of vertices. Earlier work established that, for random k$$ k $$ ‐regular graphs, the typical max‐cut value is nk/4+nP⋆k/4+on(n)+nok(k)$$ nk/4+n{P}_{star}sqrt{k/4}+{o}_n(n)+n{o}_kleft(sqrt{k}right) $$ . Therefore our algorithm is nearly optimal on such graphs. An immediate corollary of this result is that random regular graphs have nearly minimum max‐cut, and nearly maximum min‐bisection among all regular locally treelike graphs. This can be viewed as a combinatorial version of the near‐Ramanujan property of random regular graphs.
给定一个k次$$ k $$的n $$ n $$个顶点的图G $$ G $$，我们考虑在多项式时间内计算近最大切割或近最小对分的问题。对于周长2L $$ 2L $$的图，我们开发了一个复杂度为O(nkL) $$ O(nkL) $$的本地消息传递算法，该算法在所有L $$ L $$‐local算法中获得了接近最优的切值。该算法以max - cut为中心，构造了一个cut值为nk/4+nP -百科- k/4+err(n,k,L) $$ nk/4+n{P}_{star}sqrt{k/4}+mathsf{err}left(n,k,Lright) $$，其中P -百科≈0.763166 $$ {P}_{star}approx 0.763166 $$是自旋玻璃理论中的Parisi公式的值，err(n,k,L)=on(n)+nok(k)+nkoL(1) $$ mathsf{err}left(n,k,Lright)={o}_n(n)+n{o}_kleft(sqrt{k}right)+nsqrt{k}{o}_L(1) $$(下标表示渐近变量)。我们的结果推广到局部树状图，即在去除一小部分顶点后，其周长变为2L $$ 2L $$的图。早期的研究表明，对于随机k $$ k $$正则图，典型的最大切值为nk/4+nP - k/4+on(n)+nok(k) $$ nk/4+n{P}_{star}sqrt{k/4}+{o}_n(n)+n{o}_kleft(sqrt{k}right) $$。因此，我们的算法在这样的图上几乎是最优的。这个结果的一个直接推论是随机正则图在所有正则局部树状图中具有几乎最小的最大切和几乎最大的最小分。这可以看作是随机正则图的近拉马努金性质的一个组合形式。
{"title":"Local algorithms for maximum cut and minimum bisection on locally treelike regular graphs of large degree","authors":"A. Alaoui, A. Montanari, Mark Sellke","doi":"10.1002/rsa.21149","DOIUrl":"https://doi.org/10.1002/rsa.21149","url":null,"abstract":"Given a graph G$$ G $$ of degree k$$ k $$ over n$$ n $$ vertices, we consider the problem of computing a near maximum cut or a near minimum bisection in polynomial time. For graphs of girth 2L$$ 2L $$ , we develop a local message passing algorithm whose complexity is O(nkL)$$ O(nkL) $$ , and that achieves near optimal cut values among all L$$ L $$ ‐local algorithms. Focusing on max‐cut, the algorithm constructs a cut of value nk/4+nP⋆k/4+err(n,k,L)$$ nk/4+n{P}_{star}sqrt{k/4}+mathsf{err}left(n,k,Lright) $$ , where P⋆≈0.763166$$ {P}_{star}approx 0.763166 $$ is the value of the Parisi formula from spin glass theory, and err(n,k,L)=on(n)+nok(k)+nkoL(1)$$ mathsf{err}left(n,k,Lright)={o}_n(n)+n{o}_kleft(sqrt{k}right)+nsqrt{k}{o}_L(1) $$ (subscripts indicate the asymptotic variables). Our result generalizes to locally treelike graphs, that is, graphs whose girth becomes 2L$$ 2L $$ after removing a small fraction of vertices. Earlier work established that, for random k$$ k $$ ‐regular graphs, the typical max‐cut value is nk/4+nP⋆k/4+on(n)+nok(k)$$ nk/4+n{P}_{star}sqrt{k/4}+{o}_n(n)+n{o}_kleft(sqrt{k}right) $$ . Therefore our algorithm is nearly optimal on such graphs. An immediate corollary of this result is that random regular graphs have nearly minimum max‐cut, and nearly maximum min‐bisection among all regular locally treelike graphs. This can be viewed as a combinatorial version of the near‐Ramanujan property of random regular graphs.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"65 1","pages":"689 - 715"},"PeriodicalIF":1.0,"publicationDate":"2021-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82572542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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